What's a proof that the angles of a triangle add up to 180°?Beautiful. Just to emphasize what may be obvious to most, since this depends on the fifth postulate it does not hold true in non-Euclidean geometries. In spherical geometry for instance the sum of the interior angles of a triangle will always exceed 180.

Aug27

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What is a field?@celtschk This is true, but only to a degree. When I wanted to deal with a circle using a computer, I thought about pi and e, the computer worked with binary approximations that were close enough for the pixels on the screen. Of course, with packages like sympy the computer could deal with them simply as symbols without numeric values that obeyed certain rules as well, but even then it wasn't dealing with pi and e the way I was and I probably wasn't dealing with them the way a real expert would.

Gödel's incompleteness theorem can't be proven?@DougSpoonwood You have a point that we should continue to exam, and when appropriate, challenge things we think we know. However, stating that a proof is wrong is a positive claim itself, and the other side has already offered their proof in the form of their proof, they do not need to be provoked. In this particular case, the proof has been offered, and expounded upon for easier understanding, for decades. It is far more productive to ask for help understanding than to challenge something like that, at least unless and until you do understand it and have a very strong argument against it.

Gödel's incompleteness theorem can't be proven?@DougSpoonwood Who has the burden of proof depends on the context, but a useful rule of thumb is that extraordinary claims require extraordinary proof. Saying that a proof that has been accepted by mathematical community for decades with thorough examination is an extraordinary claim. That does not necessarily immediately make it false, but it does, as Asaf said originally, hint that someone should be more cautious than normal when saying that such a proof is wrong and start with the assumption that they, rather than the proof, is mistaken until they have very solid arguments.

Jan13

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Gödel's incompleteness theorem can't be proven?@DougSpoonwood Dialetheism is not exactly widely accepted in any context (and comes closest to being useful in contexts with ambiguities, such as law rather than math). But even if dialetheism has its uses in some places, dialetheism still defines a contradiction in the same way, it merely posits that there are certain systems in which a contradiction can be true.

Jan13

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Gödel's incompleteness theorem can't be proven?@DougSpoonwood In 1700 AD no one made those arguments because there were no substantial errors in Euclid's proofs. I think few people considered that there would be other geometries, but the idea that the fifth postulate did not have to be accepted like the others had been around for centruies, hence the numerous attempts to prove it. It is true that it is not a rigorous argument, as Asaf agrees, but when something is established, then it makes sense to assume that you are misunderstanding rather than everyone else being wrong, and only move to saying they are wrong with careful inspection.

Jan13

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Gödel's incompleteness theorem can't be proven?@Benny I am not quite sure what you mean in that comment. Are you essentially getting at the concept of an "undefined term"? Any complex system at its base will have some undefined terms with "point" and "line" being perhaps the best known examples. But that doesn't mean that even undefined terms do not have a meaning and it does not affect the consistency of the system in which the undefined terms are embedded.

Jan12

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Gödel's incompleteness theorem can't be proven?Consistency, at least as used in this context, is fully defined meaning essentially "free of contradiction" where contradiction is used in its formal sense of "There exists a p such that both p can be proven to be true and p can be proven to be false."

Jan3

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What is a field?@ThomasAndrews You have a great point, but I had a strong computer background and ran into Pi and e frequently, so the reals were more comfortable for me personal at that point. For many others, the rationals might be better, and they are certainly simpler.

What does increase globally by percentage mean?@DavidSchwartz I think it depends on how you interpret the statement. I think "the total sales revenue for all countries combined rises by %. " is the best interpretation and then the change in each country could be very different as you say. But I also think DiscDude's #1 interpretation is at least consistent with the wording (though unlikely). I do not think it could reasonably ever mean #2.

What does increase globally by percentage mean?As you point out, the statement is ambiguous, but I think most people would go with your alternative of the total of all trade goes up by 10%. This could permit some countries to actually have trade decline, while total trade goes up by 10%.